Problem definition

Domain

Initial Condition

Boundary condition

Exact solution

Numerical method

Space

Explicit Scheme (DRP)

The coefficients can be found in Tam(1993).At the right boundaries use fourth order central difference and fourth backward difference.At left boundaries use second order central difference for i=2 and fourth order central difference for i=3.The Dispersion relation preserving (DRP) finite volume scheme can be found in Popescu (2005).

Implicit Scheme(Compact)

Domain:

Boundaries:

where v refers to the first derivative.For a general treatment of compact scheme refer to Lele (1992).In this test case the following values are used

Both the schemes are 4th order accurate in the domain.The compact scheme has third order accuracy at the boundary.